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Asymptotics for constrained Dirichlet distributions. (English) Zbl 1329.62080
Summary: We derive the asymptotic approximation for the posterior distribution when the data are multinomial and the prior is Dirichlet conditioned on satisfying a finite set of linear equality and inequality constraints so the posterior is also Dirichlet conditioned on satisfying these same constraints. When only equality constraints are imposed, the asymptotic approximation is normal. Otherwise it is normal conditioned on satisfying the inequality constraints. In both cases the posterior is a root-$$n$$-consistent estimator of the parameter vector of the multinomial distribution. As an application we consider the constrained Polya posterior which is a non-informative stepwise Bayes posterior for finite population sampling which incorporates prior information involving auxiliary variables. The constrained Polya posterior is a root-$$n$$-consistent estimator of the population distribution, hence yields a root-$$n$$-consistent estimator of the population mean or any other differentiable function of the vector of population probabilities.
MSC:
 62E20 Asymptotic distribution theory in statistics 62D05 Sampling theory, sample surveys 62F15 Bayesian inference 62F30 Parametric inference under constraints
Software:
polyapost; R; rcdd
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